High Order Numerical Homogenization for Dissipative Ordinary Differential Equations

TL;DR

This paper introduces a high order numerical homogenization technique for dissipative ODEs with multiple time scales, improving accuracy through a refined microscopic solver within the heterogeneous multiscale method framework.

Contribution

The paper develops a high order homogenization method for dissipative ODEs using an iterative microscopic solver, enhancing accuracy beyond first order.

Findings

Achieved arbitrary order accuracy with a refined microscopic solver.

Validated the method through numerical examples.

Provided an efficient implementation for practical use.

Abstract

We propose a high order numerical homogenization method for dissipative ordinary differential equations (ODEs) containing two time scales. Essentially, only first order homogenized model globally in time can be derived. To achieve a high order method, we have to adopt a numerical approach in the framework of the heterogeneous multiscale method (HMM). By a successively refined microscopic solver, the accuracy improvement up to arbitrary order is attained providing input data smooth enough. Based on the formulation of the high order microscopic solver we derived, an iterative formula to calculate the microscopic solver is then proposed. Using the iterative formula, we develop an implementation to the method in an efficient way for practical applications. Several numerical examples are presented to validate the new models and numerical methods.